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Last updated: 30 Sep 2014

MATHEMATICS of RELATIVITY

(OBSERVER’S MATHEMATICS)

Boris Khots, Dmitriy Khots

Abstract

    Observer dependent ascending chain of embedded sets of decimal fractions and their Cartesian products is considered. For every set, arithmetic operations are defined (these operations locally coincide with standard operations), which transform every set into a local ring. The basic problems of Algebra, Geometry, Topology, and Logic are solved for this chain. Definition of Dimension of these sets is introduced. In particular, the dimension of each of these sets is greater than or equal to seven. Euclidean, Lobachevsky, and Riemannian Geometries become the particular cases of the developed Geometry, although many others are possible. For example, we proved that two lines in a plane may intersect each other in 0 (without being parallel in the usual sense), 1, 2, 10, or even 100 points. The three classical Geometries depend on on a particular neighborhood of a given line. For example, Euclidean Geometry works in sufficiently small neighborhood of the given line, but when we enlarge the neighborhood, Lobachevsky and Riemannian Geometries take over. Developed Topology gives birth to Time, and Time becomes a function of Space. Also, the Axiom of Choice becomes invalid in the new model of Mathematics. The application of the new model to Einstein's special theory of relativity is considered. The existence of the Time and Space quantums is proved. We also prove that "small" displacements may happen instantaneously, i.e. faster than the speed of light. Classical effects of "time delay", "length reduction", and "nonsimultaneous events" become invalid for "small" displacements..

News

The Math of Relativity was presented by authors on:

August 12 – 22, 2014
Presentation at the International Congress of Mathematicians 2014, "ICM2014" Seoul, South Korea
NOTE: Certain results that have been predicted by Quantum Mechanics (QM) theory are not always supported by experiments. This defines a deep crisis in contemporary physics and, in particular, quantum mechanics. We believe that, in fact, the mathematical apparatus employed within today’s physics is a possible reason. In particular, we consider the concept of infinity that exists in today mathematics as the root cause of this problem. We have created Observer’s Mathematics that offers an alternative to contemporary mathematics. This presentation is an attempt to explain how Observer’s Mathematics may explain some of the contradictions in QM theory results.

July 14 – 18, 2014
Presentations (keynote talk and section talk) at the ICNPAA 2014 conference (world congress), Narvik, Norway

June 9 – 12, 2014
Presentation at Vaxjo University International Conference "Quantum Theory from Problems to Advances - QTPA", Vaxjo, Sweden

September 9 – 12, 2013
Presentation at 8th International Conference “Topology, Geometry and Geometry teaching”, Cherkassy State Technology University, Cherkassy, Ukraine. Authors would like to thank Professor Yury Aminov for the presentation of their talk.

August 25 – 29, 2013
Presentation at the SPIE Optics + Photonics 2013 Conference “8832 - The Nature of Light: What are Photons? V”, San Diego, USA

June 10 – 13, 2013
Presentation at Vaxjo University International Conference "Quantum Theory: Advances and Problems - QTAP", Vaxjo, Sweden

June 4 – 9, 2013
Presentation at the International Conference "Nonlinear Mathematical Physics: Twenty years of JNMP", Nordfjordeid, Norway. Authors would like to thank Professor Norbert Euler for the presentation of their talk

May 25 – 27, 2013
Presentation at 10th HSTAM (The Hellenic Society for Theoretical and Applied Mechanics) 2013 International Congress on Mechanics, Chania, Crete, Greece.

June 11-14, 2012
Presentation at the Vaxjo University International Conference “Quantum Theory: Reconsideration of Foundations - 6”, Vaxjo, Sweden

June 12-15, 2012
Presentation at the 5th Chaotic Modeling and Simulation International Conference (CHAOS2012), Athens, Greece. Authors would like to thank Professor Christos H. Skiadas for the presentation of their talk.

July 23, 2012
Presentation at Kazan State University, Department of Gravity and Theory of Relativity (Prof. A. Aminova), Kazan, Russia

July 10-14, 2012
Presentation at the ICNPAA 2012 conference(world congress), Vienna, Austria

November 10-11, 2011
Presentation at “Contemporary information technologies in logistic-transportation control systems” International Conference, International Informatization Akademy, Kazan, Tatarstan, Russia. Authors would like to thank Academic Albert Bekmullin for the presentation of their talk.

August 22-25, 2011
Presentation at “The nature of light: What are Photons? IV” SPIE International Conference, San Diego, California, USA.

June 13-16, 2011
Presentation at “Foundations of Probability and Physics (FPP-6)” International Conference, Vaxjo University, Sweden.

May 29 - June 3, 2011
Presentation at “Chaotic Modeling and Simulation (CHAOS 2011)” International Conference, Agios Nikolaos, Crete Greece.

November 1-6, 2010
Presentation at "The Petrov 2010 Anniversary Symposium on General Relativity and Gravitation" at the Kazan State University, Russia

August 19-27, 2010
Presentation at the International Congress of Mathematicians 2010, "ICM2010" Hyderabad, India
NOTE: Authors showed that negative solution of the classical 10th Hilbert problem depends on validity of “Axiom of Choice” in standard (“naive”) Mathematics. In Mathematics of Relativity this axiom becomes invalid (authors proved that), and 10th Hilbert problem has positive solution

August 3-7, 2010
Presentation at "17th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications (17th ICFIDCAA)" Ho Chi Min City University of Pedagogy, Vietnam
Authors would like to thank Professor Le Hung Son for the presentation of their talk.

June 1-5, 2010
Presentation at "3rd Chaotic Modeling and Simulation” International Conference," Chania Crete, Greece

September 14-20, 2009
presentation at the Novosibirsk University The International Conference "Contemporary Analysis and Geometry"
Authors would like to thank Professor Sergey Vodopyanov for the presentation of their talk “Geometrical and Analytical aspects of Observer’s Mathematics”.

June 22-27, 2009
presentation at the International Conference “Geometry “in large”, topology and applications”, devoted to the 90th anniversary Alexey Vasilievich Pogorelov,
Kharkov, Ukraine

June 1-5, 2009
presentation at the “2nd Chaotic Modeling and Simulation” International Conference,
Chania Crete, Greece

June 14-18, 2009
presentation at the Vaxjo University International Conference “Quantum Theory: Reconsideration of Foundations - 5”,
Vaxjo, Sweden

August 24-27, 2008
presentation at Vaxjo University Conference “Foundations of Probability and Physics- 5”
Vaxjo, Sweden

August 21-26, 2008
presentation at Debrecen University 6th Bolyai – Gauss - Lobachevsky Conference (BGL6) “Non-Euclidean Geometry and its Applications”, Debrecen, Hungary

August 12-18, 2008
presentation at Fifth International Conference of Applied Mathematics and Computing, Plovdiv, Bulgaria

July 28 – August 1, 2008
presentation at 16th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications (16th ICFIDCAA), Dongguk University (Gyeongju), Korea
Authors would like to thank Professor Junesang Choi for the presentation of their talk.

June 11-16, 2007
presentation at Vaxjo University Conference “Quantum Theory: Reconsideration of Foundations - 4”,  Vaxjo, Sweden.

May 18-20, 2007
presentation at Midwest Geometry Conference – 2007,  The University of Iowa, Iowa City, Iowa, USA

October 14, 2006
presentation at
Kazan State University, Department of Gravity and Theory or Relativity (Prof. A. Aminova), Kazan, Russia

August 22-30, 2006
presentation at International Congress of Mathematicians 2006, ICM2006, Madrid, Spain
NOTE: Authors showed that negative solution of the classical Fermat problem depends on validity of “Axiom of Choice” in standard (“naive”) Mathematics. In Mathematics of Relativity this axiom becomes invalid (authors proved that), and Fermat problem has positive solution – see below.

October 6-8, 2005
presentation at Wolfram Technology Conference 2005, http://www.wolfram.com/techconf2005, Champaign, Illinois, USA

September 30, 2005
presentation at The University of Iowa, www.math.uiowa.edu, Department of Mathematics, Topology Seminar (Prof. Jon Simon), Iowa City, Iowa, USA

August 22-27, 2005
presentation at International Symposium “Analytic Function Theory, Fractional Calculus and Their Applications”, http://www.pims.math.ca/science/2005/05hms/program.html, University of Victoria, Department of Mathematics and Statistics, Victoria, Canada

June 22 – July3, 2005
presentation at the XVIIth Summer School-Seminar VOLGA-2005, www.ksu.ras.ru , Kazan, Tatarstan, Russia

April 21, 2005
presentation at The University of Iowa, www.math.uiowa.edu, Department of Mathematics, Mathematical Biology Seminar (Prof. Herbert Hethcold), Iowa City, Iowa, USA

31st January, 2005
presentation at Higher School of Economics, Department of Mathematics (Prof. S. Strunkov), Moscow, Russia

2nd December, 2004
presentation at Sibirian State Geodesics Academy, Department of Higher Mathematics (Prof. I. Vovk), Novosibirsk, Russia

1st December, 2004
presentation at Novosibirsk State University, Department of Mathematics (Prof. A. Gutman), Novosibirsk, Russia

30th November, 2004 and 3rd December, 2004
presentations at Novosibirsk State Technical University, Department of Mathematics (Prof. V. Seleznev), Novosibirsk, Russia

29th November, 2004
presentation at Novosibirsk State University, Department of  Mathematics (Prof. Sergey Krendelev), Novosibirsk, Russia

1st October, 2004
presentation at Kazan State University, Department of Gravity and Theory or Relativity (Prof. A. Aminova), Kazan, Russia

29-30 September, 2004
International Conference "Informatics problems in third millennium", Kazan, Russia

22 June - 3 July, 2004
THE XVIth SUMMER SCHOOL-SEMINAR VOLGA-2004, Russia, Tatarstan, Kazan

5 - 10 July, 2004
Conference on Non Standard Mathematics, Portugal, University of Aveiro

Mathematics of Relativity Publications

Boris Khots, Dmitriy Khots, Classical and Quantum Mechanics aspects from Observer’s Mathematics point of view, Talk at the International Congress of Mathematicians, Seoul 2014, Proceedings of ICM2014.

Dmitriy Khots, Boris Khots, What is Behind Small Deviations of Quantum Mechanics Theory from Experiments? Observer’s Mathematics Point of View, American Institute of Physics, Melville, New York (in print)

Dmitriy Khots, Boris Khots, Photoelectric Effect from Observer’s Mathematics Point of View, American Institute of Physics, Melville, New York (in print)

Boris Khots, Dmitriy Khots, Observer’s Mathematics calculations for the applications to the Quantum Mechanics, (in print).

Boris Khots, Dmitriy Khots, Quantum theory from Observer’s Mathematics point of view, 10th HSTAM 2013 Congress Proceedings, Dynamics,
#9, 6 pp, ISBN: 124-7485-789679-567-3, Technical University of Crete, Greece, 2013

Dmitriy Khots, Boris Khots, Observer’s Mathematics approach to the Quantum Mechanics, Talk at the International Conference "Nonlinear Mathematical Physics: Twenty years of JNMP", Nordfjordeid, Norway, 2013.

Boris Khots, Dmitriy Khots, Observer’s Mathematics applications to the Quantum Mechanics, Proceedings of Vaxjo University International Conference "Quantum Theory: Advances and Problems - QTAP" (in print).

Dmitriy Khots, Boris Khots, Geometrical and Quantum Mechanical aspects in Observer’s Mathematics, Proceedings of the SPIE Optics + Photonics 2013 Conference “8832 - The Nature of Light: What are Photons? V”,
vol. 8832, pp 8832 ID-1 – 8832 ID-6, San Diego, California, USA, 2013

Boris Khots, Dmitriy Khots, The foundations of Quantum Mechanics in Observer’s Mathematics, Quantum Theory: Reconsideration of Foundations - 6, AIP Conference proceedings, Volume 1508,
pp 407-411, Melville, New York, 2012

Boris Khots, Dmitriy Khots, Quantum Mechanics problems in Observer’s Mathematics, ICNPAA 2012, AIP Conference proceedings, Volume 1493,
pp 518-522, Melville, New York, 2012

Boris Khots, Dmitriy Khots, The basis of Quantum Mechanics in Observer’s Mathematics, Abstract of the talk at the 5th Chaotic Modeling and Simulation International Conference (CHAOS2012), Athens, Greece.

Boris Khots, Dmitriy Khots, Probability in Quantum Theory from Observer’s Mathematics point of view, FOUNDATIONS OF PROBABILITY AND PHYSICS - 6. AIP Conference Proceedings, Volume 1424, pp. 154-159 (2012).

Boris Khots, Dmitriy Khots, Observer’s Mathematics applications to Number Theory, Geometry, Analysis, Classical and Quantum Mechanics, Ученые Записки Казанского Университета, Серия Физико-Математические науки, 2011, Т. 153, кн. 3, С. 196-203

Boris Khots, Dmitriy Khots, Hamilton equations of general relativity in Observer’s Mathematics, Proceedings of 4th Chaotic Modeling and Simulation (CHAOS 2011) International Conference, Agios Nikolaos, Crete Greece, pp 203-208, 2011

Boris Khots, Dmitriy Khots, Observer’s Mathematics possible applications to Logistic-Transportation systems, Talk at “Contemporary information technologies in logistic-transportation control systems” International Conference, International Informatization Akademy, Kazan, Tatarstan, Russia, November 2011

Boris Khots, Dmitriy Khots, Two-slit interference and wave-particle duality for single photons from Observer’s Mathematics point of view, SPIE Proceedings, vol. 8121, H1-H6, 2011

Boris Khots, Dmitriy Khots, Analogy of Hilbert’s ten problem in Observer’s Mathematics, Talk at the International Congress of Mathematicians, Hyderabad 2010, Proceedings of ICM2010

Dmitriy Khots, Boris Khots, Lehmer’s number in Observer’s Mathematics, Talk at "17th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications (17th ICFIDCAA)", Ho Chi Min City University of Pedagogy, Vietnam, August 2010

Dmitriy Khots, Boris Khots, Chaos problems in Observer’s Mathematics, Chaos theory: Modeling, Simulations and Applications, pp 215 – 222, World Scientific, 2011

Dmitriy Khots, Boris Khots, Chaos from Observer’s Mathematics point of view, “Chaotic systems. Theory and applications”, pp 111-118, World Scientific, 2010

Dmitriy Khots, Boris Khots, Quantum Theory from Observer’s Mathematics point of view, American Institute of Physics, volume 1232, pp 294-298, Melville, New York, 2010

Dmitriy Khots, Boris Khots, Solitary Waves and Dispersive Equations from Observer’s Mathematics point of view, “Geometry “in large”, topology and applications”, pp 86-95, Kharkov, Ukraine,2010.

Dmitriy Khots, Boris Khots, Physical Aspects of Observer’s Mathematics, American Institute of Physics, volume 1101, pp 311-313, Melville, New York, 2009

Dmitriy Khots, Boris Khots, Non-Euclidean Geometry in Observer’s Mathematics, Acta Physica Debrecina, tomus XLII, pp 112-119, Debrecen, Hungary, August 2008

Dmitriy Khots, Boris Khots, Data Mining in Observer’s Mathematics, International Journal of Pure and Applied Mathematics, volume 51, #2, pp 195-201, Bulgaria, 2009

Dmitriy Khots, Boris Khots, Tenth Hilbert Problem in Observer’s Mathematics, Proceedings of the 16th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications (16th ICFIDCAA), pp 81-85, Dongguk University, Gyeongju, Korea, 2008

Dmitriy Khots, Boris Khots, Fermat’s, Mersenne’s and Waring’s problems in Observer’s Mathematics, International Journal of Pure and Applied Mathematics, Bulgaria, v. 43, #3, pp 403-408 (2008)

Dmitriy Khots, Boris Khots, Quantum Theory and Observer’s Mathematics, American Institute of Physics (AIP), volume 962, pp 261-264, 2007.

Boris Khots, Dmitriy Khots, Observer’s Mathematics – Mathematics of Relativity, Applied Mathematics and Computations, volume 187, issue 1, April 2007, pp 228-238, New York.

Boris Khots, Dmitriy Khots, An Introduction to the Mathematics of Relativity, Lecture Notes in Theoretical and Mathematical Physics, Ed. A.V. Aminova, Kazan State University, v. 7, pp 269-306, 2006.

Boris Khots, Dmitriy Khots, Physical Theory of Relativity and Mathematics of Relativity, Recent Problems in Field Theory, Ed. A.V. Aminova, Kazan State University, v. 5, pp 239-242, 2006.

Dmitriy Khots, Euclidean and Lobachevsky Geometries in Mathematics of Relativity, Recent Problems in Field Theory, Ed. A.V. Aminova, Kazan State University, v. 5, pp 243-246, 2006.

Boris Khots, Dmitriy Khots, Analogy of Fermat’s last problem in Observer’s Mathematics - Mathematics of Relativity, Talk at the International Congress of Mathematicians, Madrid 2006, Proceedings of ICM2006

 

Full text in .pdf you can see below.

An Introduction to the Mathematics of Relativity (first edition - 2004)

Title
Dedication and Thanks
AMS Classification
Chapter 0. Introduction
Chapter 1. Arithmetic
Chapter 2. Algebra
Chapter 3. Geometry
Chapter 4. Analysis & Topology
Chapter 5. Logic
Chapter 6. Einstein

Appendix 1 TABLE: V2 X2 99.99 (nonnegative elements)
Appendix 2 TABLE: V2 X2 99.98 (nonnegative elements)
Appendix 3 TABLE: V2 X2 99.97 (nonnegative elements)
Appendix 4 TABLE: V2 X2 99.95 (nonnegative elements)
Appendix 5 TABLE: V2 X2 99.92 (nonnegative elements)
Appendix 6 TABLE: V2 X2 99.90 (nonnegative elements)
Appendix 7 TABLE: V2 X2 99.53 (nonnegative elements)
Appendix 8 TABLE 1: V2 X2 99.99 +2 V2 X2 99.98
Appendix 9 TABLE 2: ((V2 X2 99.99 +2 V2 X2 99.98) +2 V2 X2 99.97) (except data from Table 1)
Appendix 10 TABLE 3: (((V2 X2 99.99 +2 V2 X2 99.98) +2 V2 X2 99.97) +2 V2 X2 99.95) (except data from Tables 1, 2)
Appendix 11 TABLE 4: ((((V2 +2 99.99 +2 V2 X2 99.98) +2 V2 X2 99.97) +2 V2 X2 99.95) +2 V2 X2 99.92) (except data from Tables 1,2,3)
Appendix 12 TABLE 5: (((((V2 X2 99.99 +2 V2 X2 99.98) +2 V2 X2 99.97) +2 V2 X2 99.95) +2 V2 X2 99.92) +2 V2 X2 99.90) (except data from Tables 1,2,3,4)
Appendix 13 TABLE 6: ((((((V2 X2 99.99 +2 V2 X2 99.98) +2 V2 X2 99.97) +2 V2 X2 99.95) +2 V2 X2 99.92) +2 V2 X2 99.90) +2 V2 X2 99.53 (except data from Tables 1,2,3,4,5,6)
Appendix 14 Intersecting Lines on Plane – Empty Intersection
Appendix 15 Intersecting Lines on Plane – Intersection number = 1
Appendix 16 Intersecting Lines on Plane – Intersection number = 2
Appendix 17 Intersecting Lines on Plane – Intersection number = 10
Appendix 18 Intersecting Lines on Plane – Intersection number = 100
Appendix 19 Riemannian Geometry: Intersecting Planes on Cube – Empty Intersection
Appendix 20 Riemannian Geometry: Intersecting Planes on Cube – Intersection number = 2
Appendix 21 Riemannian Geometry: Intersecting Planes on Cube – Intersection number = 4
Appendix 22 Riemannian Geometry: Intersecting Planes on Cube – Intersection number = 20
Appendix 23 Appendixes 14-22, Summary
 

© 2004 Boris Khots, Dmitriy Khots